This completes the proof of Jensen’s Coding Theorem, subject to theverification of Relativized □and ⋄.Section Four Relatived Square and DiamondFor completeness, we prove Relativized □and ⋄. As relativization causes no se-rious problems, we first establish unrelativized versions, and then afterward indicatewhat modifications are required.

We begin with □.First we prove □in the following form:Global □Assume V = L. Then there exists ⟨Cµ|µ a singular limit ordinal⟩suchthat:(a) Cµ is CUB in µ(b) ordertype (Cµ) < µ(c) ¯µ ∈Lim Cµ −→C¯µ = C∩¯µ.In the proof we shall take advantage of Jensen’s Σ∗theory, as reformulated inFriedman [94]. For the convenience of the reader we describe that theory here.For simplicity of notation, for limit ordinals µ we let eJµ denote Jα whereωα = µ.

So ORD( eJµ) = µ.Let M denote some Jα, α > 0. (More generally, our theory applies to “accept-able J-models”.) We make the following definitions, inductively.

We order finitesets of ordinals by the maximum difference order: x < y iffα ∈Y where α is thelargest element of (y −x) ∪(x −y).1) A Σ∗1 formula is just a Σ1 formula. A predicate is Σ∗1 (Σ∗1, respectively) ifit is definable by a Σ∗1 formula with (without, respectively) parameters.

ρM1= Σ∗1projectum of M = least ρ s.t. there is a Σ∗1 subset of ωρ not in M and pM1=least p s.t.

A ∩ρM1/∈M for some A Σ∗1 in parameter p (where p is a finite set ofordinals). HM1= HMωρM1 = sets x in M s.t.

M-card (transitive closure (x)) < ωρM1 .For any x ∈M, M1(x) = First reduct of M relative to x = ⟨HM1 , A1(x)⟩whereA1(x) ⊆HM1codes the Σ∗1 theory of M with parameters from HM1∪{x} in thenatural way: A1(x) = {⟨y, n⟩| the nth Σ∗1 formula is true at ⟨y, x⟩, y ∈HM1 }. Agood Σ∗1 function is just a Σ1 function and for any X ⊆M the Σ∗1 hull (X) is justthe Σ1 hull of X.13

(2) For n ≥1, a Σ∗n+1 formula is one of the form ϕ(x) ←→Mn(x) |= ψ, whereψ is Σ1. A predicate is Σ∗n+1 (Σ∗n+1, respectively) if it is defined by a Σ∗n+1 formulawith (without, respectively) parameters.

ρMn+1 = Σ∗n+1 projectum of M = least ρsuch that there is a Σ∗n+1 subset of ωρ not in M and pMn+1 = pMn ∪p where p is leastsuch that A ∩ρMn+1 /∈M for some A Σ∗n+1 in parameter pMn ∪p. HMn+1 = HMωρMn+1 =sets x in M s.t.

M-card (transitive closure (x)) < ωρMn+1. For any x ∈M, Mn+1(x) =(n + 1) st reduct of M relative to x = ⟨HMn+1, An+1(x)⟩where An+1(x) ⊆HMn+1codes the Σ∗n+1 theory of M with parameters from HMn+1 ∪{x} in the natural way:An+1(x) = {⟨y, m⟩| the mth Σ∗n+1 formula is true at ⟨y, x⟩, y ∈HMn+1}.

A good Σ∗n+1function f is a function whose graph is Σ∗n+1 with the additional property that forx ∈Dom(f), f(x) ∈Σ∗n hull (HMn ∪{x}). The Σ∗n+1 hull (X) for X ⊆M is theclosure of X under good Σ∗n+1 functions.Facts.

(a) ϕ, ψΣ∗n formulas −→ϕ ∨ψ, ϕ ∧ψ are Σ∗n formulas(b) ϕΣ∗n or Q∗n (= negation of Σ∗n) −→ϕ is Σ∗n+1(c) Y ⊆Σ∗n hull (X) −→Σ∗n hull (Y ) ⊆Σ∗n hull (X)(d) f good Σ∗n function −→f good Σ∗n+1 function(e) Σ∗n hull (X) ⊆Σ∗n+1 hull (X)(f)There is a Σ∗n relation W(e, x) s.t. if S(x) is Σ∗n then for some e ∈ω,S(x) ←→W(e, x) for all x.

(g) The structure Mn(x) = ⟨HMn , An(x)⟩is amenable. (h) HMn = JAnωρMn where An = An(0).

(i) Suppose H ⊆M is closed under good Σ∗n functions and π : M −→M,M transitive, Range (π) = H and pMn−1 ∈H (if n > 1). Then π preserves Σ∗nformulas: for Σ∗nϕ and x ∈M, M |= ϕ(x) ←→M |= ϕ(π(x)).

And (for n > 1),π(p ¯Mn−1) = pMn−1.Proof of (i)Note that H ∩Mn−1(π(x)) is Σ1-elementary in Mn−1(π(x)). Andπ−1[H ∩Mn−1(π(x))] = ⟨JAωρ, A(x)⟩for some ρ, A, A(x).

But (by induction on n)A = AMn−1 ∩JAωρ, A(x) = An−1(x) ¯M ∩JAωρ. And ρ = ρ ¯Mn−1 using our assumptionabout the parameter pMn−1.

And π−1(pMn−1) = ¯p must be p ¯Mn−1 as ¯M = Σ∗n−1 hull ofH ¯Mn−1 ∪{p ¯Mn−1}.⊣Theorem 4.1. By induction on n > 0 :14

1) If ϕ(x, y) is Σ∗n then ∃y ∈Σ∗n−1 hull (HMn−1 ∪{x})ϕ(x, y) is also Σ∗n.2)If ϕ(x1 · · · xk) is Σ∗m, m ≥n and f1(x), · · · , fk(x) are good Σ∗n functions,then ϕ(f1(x) · · ·fk(x)) is Σ∗m.3) The domain of a good Σ∗n function is Σ∗n4) Good Σ∗n functions are closed under composition.5) (Σ∗n Uniformization) If R(x, y) is Σ∗n then there is a good Σ∗n function f(x)s.t. x ∈Dom(f) ←→∃y ∈Σ∗n−1 hull (HMn−1 ∪{x})R(x, y) ←→R(x, f(x)).6)There is a good Σ∗n function hn(e, x) s.t.

for each x, Σ∗n hull ({x}) ={hn(e, x)|e ∈ω}.Proof. The base case n = 1 is easy (take Σ∗0 hull (X) = M for all X).

Now we proveit for n > 1, assuming the result for smaller n.1) Write ∃y ∈Σ∗n−1 hull (HMn−1∪{x})ϕ(x, y) as ∃¯y ∈HMn−1ϕ(x, hn−1(e, ⟨x, ¯y⟩))using 6) for n −1. Since hn−1 is good Σ∗n−1 we can apply 2) for n −1 to concludethat ϕ(x, hn−1(e, ⟨x, ¯y⟩)) is Σ∗n.

Since the quantifiers ∃e∃¯y ∈HMn−1 range over HMn−1they preserve Σ∗n-ness.2)ϕ(f1(x) · · ·fk(x)) ←→∃x1 · · ·xk ∈Σ∗n−1 hull (HMn−1 ∪{x}) [xi = fi(x)for 1 ≤i ≤k ∧ϕ(x1 · · · xk)]. If m = n then this is Σ∗n by 1).

If m > n thenreason as follows: the result for m = n implies that An(⟨f1(x) · · ·fk(x)⟩) is ∆1 overMn+1(x). Thus Am−1(⟨f1(x) · · ·fk(x)⟩) is ∆1 over Mm−1(x).

So as ϕ is Σ∗m we getthat ϕ(f1(x) · · ·fk(x)) is also Σ1 over Mm−1(x), hence Σ∗m.3) If f(x) is good Σ∗n then dom(f) = {x|∃y ∈Σ∗n−1 hull of HMn−1 ∪{x}(y =f(x))} is Σ∗n by 1).4) If f, g are good Σ∗n then the graph of f ◦g is Σ∗n by 2). And f ◦g(x) ∈Σ∗n−1hull(HMn−1 ∪{x}) since the latter hull contains g(x), f is good Σ∗n and Fact c) holds.5) Using 6) for n −1, let R(x, ¯y) ←→R(x, hn−1(¯y)) ∧¯y ∈HMn−1.

Then R is Σ∗nby 2) for n −1 and using Σ1 uniformization on (n −1) s.t. reducts we can define agood Σ∗n function ¯f s.t.

R(x, ¯f(x)) ←→∃¯y ∈HMn−1R(x, ¯y). Let f(x) = hn−1( ¯f(x)).Then f is good Σ∗n by 4).6) Let W be universal Σ∗n as in Fact f).

By 5) there is a good Σ∗n g(e, x) s.t.∃y ∈Σ∗n−1 hull(HMn−1 ∪{x}) W(e, ⟨x, y⟩) ←→W(e, ⟨x, g(e, x)⟩) (and g(e, x) defined−→W(e, ⟨x, g(e, x)⟩)). Let hn(e, x) = g(e, x).

If y ∈Σ∗n hull ({x}) then for somee, W(e, ⟨x, y′⟩) ←→y′ = y so y = hn(e, x). Clearly hn(e, x) ∈Σ∗n hull ({x}) since15

hn is good Σ∗n.⊣Now we are ready to prove Global □. Assume V = L and let µ be a singularlimit ordinal.

Our goal is to define Cµ, a CUB subset of µ. Let β(µ) ≥µ be the leastlimit ordinal β such that µ is not regular with respect to eJβ-definable functions, andlet n(µ) be least so that there is a good Σ∗n(µ)( eJβ(µ)) partial function from an ordinalless than µ cofinally into µ.

Note that ρβ(µ)n(µ) ≤µ as otherwise such a partial functionwould belong to eJβ(µ), contradicting the leastness of β(µ). Also µ ≤ρβ(µ)n(µ)−1, elsewe have contradicted the leastness of n(µ).For X ⊆eJβ(µ) let H(X) = Σ∗n(µ) hull of X in eJβ(µ).

For some least parameterq(µ) ∈eJβ(µ), H(µ∪{q(µ)}) = eJβ(µ). (“Least” refers to the canonical well-ordering ofL.) Also let α(µ) = S{α < µ|α = H(α∪{q(µ)})∩µ}.

Then (unless α(µ) = S ∅= 0)α(µ) = H(α(µ) ∪{q(µ)}) ∩µ and α(µ) < µ. To see the latter note that for largeenough α < µ, H(α ∪{q(µ)}) contains both the domain and defining parameter fora good Σ∗n(µ) partial function from an ordinal less than µ cofinally into µ.If µ < β(µ) let p(µ) = ⟨q(µ), µ, α(µ)⟩and if µ = β(µ) let p(µ) = α(µ).We are ready to define Cµ.

Let C0µ = {¯µ < µ| For some α, ¯µ = S(H(α ∪{p(µ)}) ∩µ)}. Then C0µ is a closed subset of µ.

If C0µ is unbounded in µ thenset Cµ = C0µ. If C0µ is bounded but nonempty then let µ0 = S C0µ and defineC1µ = {¯µ < µ| For some α, ¯µ = S(H(α ∪{p(µ), µ0}) ∩µ)}.

If C1µ is unboundedthen set Cµ = C1µ. If C1µ is bounded but nonempty then let µ1 = S C1µ and defineC2µ = {¯µ < µ| For some α, ¯µ = S(H(α ∪{p(µ), µ0, µ1}) ∩µ)}.

Continue in thisway, defining Ckµ for k ∈ω until Ckµ is unbounded or empty. Note that α0 > α1 >· · · where αk is greatest so that S(H(αk ∪{p(µ), µ0, · · · , µk−1}) ∩µ) = µk, sinceαk ∈H(αk ∪{p(µ), µ0, · · · , µk−1, µk}).

So for some least k(µ) ∈ω, Ck(µ)µis indeedunbounded or empty. If Ck(µ)µis unbounded then set Cµ = Ck(µ)µ.If Ck(µ)µ= ∅then we choose Cµ to be an ω-sequence cofinal in µ, codingapproximations to the structure eJβ(µ), as follows.

(This is necessary to establishRelativized □(c).) Note that H = H({p(µ), µ0, · · · , µk(µ)−1}) is cofinal in µ sinceCk(µ)µ= ∅.

Assume first that n(µ) = 1, when Cµ is more easily described. ThenH is also cofinal in β(µ), else H ∈eJβ(µ) and µ is singular inside eJβ(µ).

Let h =h1(e, x) be the canonical good Σ∗1 Skolem function for eJβ(µ), so H = {h(e, p)|e ∈ω}where p = {p(µ), µ0, · · · , µk(µ)−1}. Let ¯σn = max({h(e, p)|e < n} ∩µ) and σn =16

max({h(e, p)|e < n} ∩β(µ)). Then Cµ = {δ0, δ1, · · ·} where δn is an ordinal codingTC(Σ∗1 hull (¯σn ∪{p}) restricted to σn), where TC denotes “transitive collapse”.By the Σ∗1 hull of X restricted to σn we mean the closure of X under hδn, obtainedby interpreting the Σ∗1 definition of h in eJσn.Now suppose n(µ) > 1, Ck(µ)µ= ∅.

Then if ρ(µ) denotes ρµn(µ)−1, H is cofinalin ρ(µ), else H ∈eJρ(µ) and µ is singular in eJρ(µ). Let h be the canonical goodΣ∗n(µ) Skolem function for eJρ(µ) and let p = {p(µ), µ0, · · · , µk(µ)−1}.

Let ¯σn =max({h(e, p)|e < n}∩µ), σn = max({h(e, p)|e < n}∩ρ(µ)). Then Cµ = {δ0, δ1, · · ·}where δn is an ordinal coding TC(Σ∗n(µ) hull (¯σn ∪{p}) restricted to σn).

The Σ∗n(µ)hull of X restricted to σn is the closure of X under hσn, obtained by replacing the(n(µ) −1) st reduct Mn(µ)−1(x) by Mn(µ)−1(x) ↾σn in the Σ∗n(µ) definition of h.(Recall that Mn(x) = ⟨JAnωρMn , An(x)⟩; by Mn(x) ↾σ we mean ⟨eJAnσ , An(x) ∩eJAnσ ⟩. )Clearly Cµ is CUB in µ, and by the same argument used to justify α(µ) < µ,the ordertype of Cµ is less than µ.

(These facts are obvious when Ck(µ)µ= ∅.) So toprove Global □we only need to check coherence: ¯µ ∈Lim Cµ −→C¯µ = Cµ ∩¯µ.Lemma 4.2.

¯µ ∈Ckµ −→Ck¯µ = Ckµ ∩¯µ.Proof. First suppose that k = 0.

Given ¯µ ∈C0µ we can choose α < ¯µ such that¯µ = S(H(α∪{p(µ)})∩µ), where H is the operation of taking the Σ∗n(µ) hull. Also letρ = S(H(α∪{p(µ)})∩ρµn(µ)−1).

Let π : eJ ¯β −→eJβ(µ) be the inverse to the transitivecollapse of H = Σ∗n(µ) hull (¯µ∪{p(µ)}) restricted to ρ. Note that any x ∈H belongsto H(µ′, ρ′) = Σ∗n(µ) hull (µ′ ∪{p(µ)}) restricted to ρ′, for some µ′ < ¯µ, ρ′ < ρ andµ′, ρ′ can be chosen to be in H. It follows that H ∩µ = ¯µ and therefore whenµ < β(µ), π(¯µ) = µ.

Also note that Σ∗n(µ)−1 hull (ρ ∪{p(µ)}) ∩ρµn(µ)−1 = ρ,so H is closed under good Σ∗n−1 functions. It follows that π :eJ ¯β −→eJβ(µ) isΣ∗n−1-elementary and ¯µ is Σ∗n(µ)−1( eJ ¯β)-regular, Σ∗n(µ)( eJ ¯β)-singular.

So ¯β = β(¯µ),n(µ) = n(¯µ). Also π(q(¯µ)) = q(µ).

Since α(¯µ) < α it must be that α(¯µ) = α(µ). Soπ(p(¯η)) = p(µ).

Now it is easy to see that C0¯µ = C0µ ∩¯µ.Now suppose k = 1. The above argument shows that ¯µ ∈C1µ −→C0¯µ = C0µ ∩¯µand hence, since µ0 < ¯µ, ¯µ0 = µ0.

Now again, the above argument shows thatC1¯µ = C1µ ∩¯µ. The general case k ≥0 now follows similarly.⊣Coherence now follows easily: if ¯µ ∈Lim Cµ and Cµ = Ckµ then by Lemma 4.2,Ck¯µ = Ckµ ∩¯µ is unbounded in ¯µ so C¯µ = Ck¯µ and we’re done.

If Ckµ = ∅for some k17

then lim Cµ = ∅so coherence is vacuous.To establish the appropriate relativized form of □we need:Lemma 4.3. Suppose π :⟨eJ¯µ, ¯C⟩Σ1−→⟨eJµ, Cµ⟩.

Then C = C¯µ and π extendsuniquely to a Σ∗n(µ)-elementary ˜π :eJβ(¯µ) −→eJβ(µ) such that p(µ) ∈Rng ˜π.Proof. First suppose that Cµ = Ckµ for some k. For µ′ ∈Cµ form H(µ′) as Hwas formed in the proof of Lemma 2 for ¯µ.

Then π(µ′) :eJβ(µ′) −→eJβ(µ) withrange H(µ′) is Σ∗n(µ)−1-elementary and eJβ(µ) = S{H(µ′)|µ′ ∈Cµ}. And π(µ′) ↾µ′ = id ↾µ′, π(µ′)(p(µ′)) = p(µ).

Now let X = Range(π) and form eX = Σ∗n(µ)hull (X ∪{p(µ)}) in eJβ(µ). If y ∈eX then for some µ′ ∈Cµ, y = π(µ′)(y′) wherey′ ∈Σ∗n(µ) hull ((X ∩eJµ′) ∪{p(µ′)}).

In particular if y ∈eJµ then y ∈Σ∗1 hull (X)in ⟨eJµ, Cµ⟩= X. So the inverse to the transitive collapse of eX = ˜π is a Σ∗n(µ)-elementary embedding extending π, with p(µ) in its range.

If ˜π :eJ ¯β −→eJβ(µ)then ¯µ = ˜π−1(µ) is singular via a Σ∗n(µ)( eJ ¯β) partial function since either Σ∗n(µ) hullof µ′ ∪{p(µ)} in eJβ(µ) is unbounded in µ for some µ′ < S(Rng(π) ∩µ), in whichcase we can assume µ′ ∈Rng π and by Σ∗n(µ)-elementary of ˜π we’re done, or if notµ∗= µ∩Σ∗n(µ) hull (µ∗∪{p(µ)}) in eJβ(µ) where µ∗= S(Rng π∩µ), contradicting thedefinition of α(µ). Since ¯µ is Σ∗n(µ)−1( eJ ¯β)-regular, we get ¯β = β(¯µ), n(µ) = n(¯µ).Then the Σ∗n(µ)-elementarity of ˜π guarantees that C = C¯µ.

The uniqueness of ˜πfollows from the fact that eJβ(¯µ) = Σ∗n(µ) hull (¯µ ∪{p(¯µ)}) and ˜π ↾¯µ is determinedby π.If Ckµ = ∅for some k then Cµ was defined as a special ω-sequence cofinal in µ.That definition was made precisely to enable the preceding argument to also applyin this case.⊣Relativized □Let S = S{Sα|α an infinite cardinal}. There exists ⟨Cs|s ∈S⟩such that Cs ∈As and:(a) Cs is closed, unbounded in µ0s, ordertype (Cs) ≤α(s).If |s| is a successor ordinal then ordertype (Cs) = ω.

(b) ν ∈Lim(Cs) −→for some η < |s|, ν = µ0s↾η and Cs ∩ν = Cs↾η. (c)Let π :⟨A, C⟩Σ1−→⟨A0s, Cs⟩and write crit(π) = ¯α, A = L¯µ[A, ¯s∗].

Ifπ(¯α) = α(s) then L[A, ¯s∗] ⊨|¯s| is not a cardinal > ¯α and(c1) C ∈Lµ[A, ¯s∗] where µ is the least p.r. closed ordinal greater than ¯µ s.t.18

Lµ[A, ¯s∗] ⊨card(|¯s|) ≤¯α. (c2) π extends to π′ : A′Σ1−→A′s where A′ = Lµ′[A, ¯s∗], µ′ = largest ordinaleither equal to ¯µ or s.t.

ω · µ′ = µ′ and Lµ′[A, ¯s∗] ⊨|¯s| is a cardinal greater than ¯α. (c3) If ¯α is a cardinal and π(¯α) = α then A = A0¯s and C = C¯s.Relativized ⋄Let E = all s ∈S such that ordertype (Cs) = ω.

There exists⟨Ds|s ∈E⟩such that Ds ⊆A0s and:(a)D ∈bAs ̸= A0s, D ⊆A0s −→{ξ < |s|s ↾ξ ∈E, Ds↾ξ = D ∩A0s↾ξ} isstationary in bAs. (b) Ds is uniformly Σ1-definable as an element of A′s.

(c) If A′s ⊨α++ exists then Ds = ∅.We now make the necessary modifications to obtain Relativized □. First, if µis a singular limit ordinal and eJµ |= α is the largest cardinal then we thin out Cµto give it ordertype ≤α : By induction on limit ¯µ ≤µ define C∗¯µ as follows.

For¯µ ≤α, C∗¯µ = ¯µ. Otherwise C∗¯µ = {ith element of C¯µ|i ∈C∗¯µ0 where ¯µ0 = ordertype(C¯µ)}.

This defines C∗µ. It is easily verified that the C∗µ enjoy all the properties ofthe Cµ except they are only defined when µ is a singular limit ordinal such thateJµ |= There is a largest cardinal.

In addition, ordertype C∗µ ≤α(µ), the largestcardinal of eJµ.Now suppose V = L, α is a cardinal, s ∈Sα, |s| > α and s is a 0-string, meaningthat s(µ) = 0 for all η ∈Dom(s). Then we can choose our predicate A = ∅, defineCs = C∗µ0s and Relativized □will hold for such 0-strings.

The final comment is thatall we have done will relativize to arbitrary strings s ∈Sα, defined relative to anarbitrary predicate A ⊆ORD, Hα = Lα[A] for all cardinals αNow we turn to Relativized ⋄. Again we begin with a nonrelativized version.Let α be a cardinal and assume V = L.⋄on α+ Let E = all µ < α+ s.t.

Cµ has ordertype ω. There exists ⟨Dµ|µ ∈E⟩s.t.Dµ ⊆eJµ and:(a) If D ⊆eJα+ then {µ ∈E|D ∩eJµ = Dµ} is stationary in α+.

(b) Dµ is uniformly Σ1 definable as an element of eJβ′(µ) where β′(µ) = largestβ s.t. either β = µ or ωβ = β and eJβ |= µ is a cardinal greater than α.

(c) If eJβ′(µ) |= α++ exists then Dµ = ∅.19

Proof. For µ ∈E let Dµ = ∅if eJβ′(µ) |= α++ exists and otherwise let ⟨Dµ, Fµ⟩beleast in eJβ′(µ) such that Fµ is CUB in µ and ¯µ ∈Fµ −→¯µ /∈E or D¯µ ̸= Dµ ∩eJ¯µ.If ⟨Dµ, Fµ⟩doesn’t exist let Dµ = ∅.

Properties (b), (c) are clear. To prove (a),suppose it fails and let ⟨D, F⟩be least in eJα++ such that D ⊆eJα+, F is CUB inα+ and µ ∈F −→µ /∈E or Dµ ̸= D ∩eJµ.

Let σ be least such that ωσ = σand ⟨D, F⟩∈eJσ. Then eJσ |= α+ is the largest cardinal.

Let H = Σ1 Skolem hullof {α+} in eJσ and µ = S(H ∩α+). Then eJβ′(µ) is the transitive collapse of theΣ1 Skolem hull of µ ∪{α+} in eJσ; let π :eJβ′(µ) −→eJσ have range equal to thelatter hull.

Then we have a contradiction provided µ ∈E. But the fact that H isunbounded in µ implies that C0µ = ∅so ordertype (Cµ) is ω and µ ∈E.⊣Now as with Relativized □, if V = L and α is a cardinal, s ∈Sα is a 0-string inE, |s| > α then we can choose A = ∅and define Ds = Dµ0s where the latter comesform ⋄on α+.

Finally, relativize everything to arbitrary reshaped strings s ∈Sαand an arbitrary predicate A ⊆ORD, Lα[A] = Hα for all cardinals α.This completes the proof of Relativized □and ⋄.ReferencesBeller-Jensen-Welch [82] Coding the Universe, Cambridge University Press.Friedman [87] Strong Coding, Annals of Pure and Applied Logic.Friedman [94] Jensen’s Σ∗Theory and the Combinatorial Content of V = L, toappear, Journal of Symbolic Logic.Jensen [72] The Fine Structure of the Constructible Hierarchy, Annals of Mathe-matical Logic.20

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